CSIR NET Maths previous year question papers

Consider a Markov chain {$X_{n} : n \ge 1$} on state space {1, 2, 3, 4, 5} with the transition probability matrix whose rows are (0, 1/2, 1/2, 0, 0), (0, 0, 1, 0, 0), (0, 1/3, 0, 1/3, 1/3), (1, 0, 0, 0, 0) and (0, 0, 0, 0, 1). Then, which of the following statements are true?

Consider the M/M/1 queue in which customers arrive according to a Poisson process with rate 3 and successive service times are independent exponential random variables having mean 1/9. Let $P_{n}$ be the long run probability that there are exactly n customers in the system. Then, which of the following statements are true?

Let X and Y be independent Poisson random variables with means 4 and 2, respectively. Then, which of the following statements are true?

A system has two components $C_{1}$ and $C_{2}$ put in parallel. The components $C_{1}$ and $C_{2}$ have independent lifetimes $X_{1}$ and $X_{2}$, respectively. The probability distribution of X_j is exponential with mean 1/j, j = 1, 2. Suppose that R(t) and h(t) are the reliability and the hazard rate functions of the system, respectively. Then, which of the following statements are true?

Let X and Y be independent and identically distributed N(0, 1) random variables. Let $S = X^{2} + Y^{2}$ and T = $e^{-(X^{2}+Y^{2})/2}$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed random variables with the probability density function f(x|$\theta) =$ $e^{-(x-\theta)}$ if $x \ge \theta$, and 0 otherwise, where $\theta \in \mathbb{R}$ is the unknown parameter. Further, let $Y = \min(X_{1}, X_{2}, \dots, X_{n})$. Which of the following are confidence intervals for $\theta$ with the confidence coefficient $(1 - \alpha)$, where $\alpha \in (0, 1)$?

Consider a paired data $(x_{i}, y_{i}); i = 1, 2, 3, 4, 5$, where $(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (-2, -1, 0, 1, 2)$ and $y_{i} = x_{i}^{2}$ for all i = 1, 2, 3, 4, 5. On this data, a simple linear regression model with an intercept term and a simple linear regression model without an intercept term are fitted using the method of least squares. Which of the following statements are true?

An analyst fits a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}X_{1i} + \dots + \beta_{4}X_{4i} + \varepsilon_{i}, i = 1, 2, \dots, n$, using the method of least squares. However, his coordinator insists that the intercept and two regressors $Z_{1} = X_{1} + X_{3}$ and $Z_{2} = X_{2} - X_{4}$ are enough to represent the model. Suppose the coordinator's claim can be tested in the form of a general linear hypothesis, viz., $H_{0} : L^{T}\beta = 0$ against $H_A : H_{0}$ is not true, where $\beta^{T} = (\beta_{0}, \beta_{1}, \dots, \beta_{4})$. Suppose we have n observations on the response Y and each regressor. Further, assume that the errors with or without restrictions are independent $N(0, \sigma^{2})$ variables. Then, which of the following statements are true?

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{10}, Y_{10})$ be a random sample from a bivariate normal distribution BVN$(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho)$ with $\mu_{1} = 5, \mu_{2} = 6, \sigma_{1}^{2} = 4, \sigma_{2}^{2} = 9$ and $\rho = 1/2$. Then, which of the following statements are true?

Let the random vector $(X_{1}, X_{2}, X_{3})^{T}$ have the positive definite dispersion matrix with rows $(1, \rho, \rho), (\rho, 1, \rho)$ and $(\rho, \rho, 1)$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots$ be a sequence of independent and identically distributed random variables with $E(X_{1}) = 0, E(X_{1}^{2}) = 1, E(X_{1}^{3}) = 0, E(X_{1}^{4}) = 3$. Let $S_{n} = \sum_{i}$₌$_{1}^{n} X_{i}, T_{n} = \sum_{i}$₌$_{1}^{n} X_{i}^{2}, U_{n} = \sum_{i}$₌$_{1}^{n} X_{i}^{3}$ and $V_{n} = \sum_{i}$₌$_{1}^{n} X_{i}^{4}$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 3)$ be a random sample from the uniform distribution on the interval $(\theta_{1} - \theta_{2}, \theta_{1} + \theta_{2})$, where $\theta_{1} \in \mathbb{R}$ and $\theta_{2} > 0$ are unknown parameters. Let $X_{(j)}$ be the jᵗʰ order statistic, j = 1, 2, …, n, and let ${\bar{X}} = (1/n)\sum_{i}$₌$_{1}^{n} X_{i}$. Here, $(X_{(1)}, X_{(n)})$ is a complete and sufficient statistic for $(\theta_{1}, \theta_{2})$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 2)$ be a random sample from a continuous distribution with the probability density function f(x|$\theta) =$ $(1/2\theta)e^{-|x-\theta|/\theta}$, $x \in \mathbb{R}$, where $\theta (> 0)$ is an unknown parameter. Let $U_{n} = (1/n)\sum_{i}$₌$_{1}^{n} X_{i}, V_{n} = (1/n)\sum_{i}$₌$_{1}^{n} X_{i}^{2}$, and $S_{n}^{2} = (1/(n-1))\sum_{i}$₌$_{1}^{n}(X_{i} - U_{n})^{2}$. Then, which of the following statements are true?

Let X be a random sample of size one from the probability density function f(x|$\theta) =$ $\theta{}e^{-\theta(x-1)}$ if x > 1, and 0 otherwise, where $\theta (> 0)$ is the unknown parameter. Suppose we want to test the null hypothesis $H_{0} : \theta = 1$ against the alternative hypothesis $H_{1} : \theta \ne 1$, based on the observed value x of X. Then, which of the following statements are true?

Let $(X_{1}, X_{2}, \dots, X_{7})$ and $(Y_{1}, Y_{2}, \dots, Y_{9})$ be two independent random samples from the continuous distribution functions $F(x - \mu)$ and $F(x - \theta)$, respectively, where $F, \mu$ and $\theta$ are all unknown. Further, let $\mu$ be the unique median of $F(x - \mu)$ and $\theta$ be the unique median of $F(x - \theta)$. Let $R_{i}$ be the rank of $Y_{i}$ in the combined sample, i = 1, 2, …, 9. For testing $H_{0} : \mu = \theta$ against $H_{1} : \mu > \theta$, the test statistic $T = \sum_{i}$₌$_{1}^{9} R_{i}$ is proposed. Then, which of the following statements are true?

If the incidence matrix of a block design is given by N with rows (1, 1, 1, 0), (1, 1, 0, 1), (1, 0, 1, 1) and (0, 1, 1, 1), then which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{15}$ be a random sample from an Exponential distribution with the probability density function f(x∣$\sigma) = (1/\sigma)\exp(-x/\sigma)$ if x>0, 0 elsewhere, where the unknown parameter $\sigma$ is positive. Let ${\bar{X}} = (1/15)\sum_{1}^{15}X_{i}$. Suppose that $\varphi$ denotes the likelihood ratio test for testing $H_{0}: \sigma\le1$ against $H_{1}: \sigma>1$ at level $\alpha=0.1$. It is given that $\chi^{2}_{15},_{0}._{1}=22.307, \chi^{2}_{15},_{0}._{9}=8.547, \chi^{2}_{30},_{0}._{1}=40.256, \chi^{2}_{30},_{0}._{9}=20.599$, where $P(W>\chi^{2}_{n},_{a})=\alpha$ and W~$\chi^{2}_{n}$. Then which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n} (n\ge2)$ be a random sample from an Exponential distribution with the probability density function f(x∣$\mu,\sigma) = (1/\sigma)\exp((\mu-x)/\sigma)$ if $x>\mu, 0$ otherwise, where parameters $\mu$ and $\sigma$ are unknown and positive. Let ${\bar{X}}_{n}, S_{n}^{2}$ and $X_{1}:_{n}$ denote the sample mean, the sample variance and the sample smallest order statistic, respectively, and let $\theta = \sigma/\mu$. Then which of the following statements are true?

Consider a dataset of n observations given by A = {$x_{1},x_{2},\dots,x_{n}$}. Create a new dataset, given by B = {$x_{1},x_{2},\dots,x_{n},-x_{1},-x_{2},\dots,-x_{n}$}. Which of the following statements are always true? (Variance is calculated with divisor as the number of observations in the corresponding dataset.)

Let $X_{1}, X_{2}, \dots, X_{n} (n\ge2)$ be a random sample from Uniform$[-3\theta/2, \theta/2]$ distribution, where $\theta>0$ is an unknown parameter. Let ${\bar{X}}, X_{(1)}$ and $X_{(n)}$ denote respectively the mean, the smallest order statistic and the largest order statistic of the sample. Then which of the following statements are true?

Suppose X∣$\theta$ ~ Binomial$(7,\theta), 0<\theta<1$, and the prior distribution of $\theta$ is Beta$(\alpha,\beta)$ where $\alpha>0$ and $\beta>0$ are known. Then which of the following statements MAY NOT be true?

Consider an M/M/2 queuing system with the birth rate $\lambda=4$ per minute, the death rate $\mu=1$ per minute, and the total capacity of 3 customers (including the ones that are being served). Let $p_{0}$ and $p_{3}$ denote the long-run probabilities that the system will be empty (i.e. without customers) and will be blocked (i.e. full), respectively. Which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{20}$ be a random sample from $N_{12}(\mu,\sum)$, where $\mu\in\mathbb{R}^{12}$ and $\sum$ is positive definite. Suppose ${\bar{X}} = (1/20)\sum_{i}$₌$_{1}^{20}X_{i}$ and $S = (1/19)\sum_{i}$₌$_{1}^{20}(X_{i}-{\bar{X}})(X_{i}-{\bar{X}})^{T}$. Then which of the following statements are true?

Consider the following $2^{5}-$factorial design with 8 blocks (Block 1: (1), acd, bce, abde; Block 2: e, acde, bc, abd; and six further blocks). Which of the following statements are true?

Let {$X_{n} : n\ge1$} be a sequence of independent and identically distributed random variables, where $X_{1}$ has an Exponential distribution with mean 1. Define $T_{n} = \max${$X_{1},X_{2},\dots,X_{n}$} − ln n, n≥1. Suppose $T_{n} \to$ᵈ Y as $n\to\infty$. Then which of the following statements are true?